Nondimensionalization and Scaling

Hi all.
To me, scaling means adopting properly scales in nondimensionalization.
However, as I see in the book "A modern introduction to the mathematical theory of water waves" by R.S.Johnson, the author distinguish the two processes in a way that confuses me much. (Sec. 1.3.1 and 1.3.2)
Can someone who have used this book kindly help clarify the two things?

I am not familiar with the book, but I believe non-dimensionalization literally gets rid of all dimensional quantities. So, you might write something like
x=Lx* where L is a typical length scale in the x direction and resplacing all your x' s by x* 's where x* is a non dimesional quantity since you have taken 10metres, say, and divided it by metres to just get 10.

Scaling on the other hand helps you get an idea of the relative sizes of terms so if you had something long in the x direction and short in the y direction you would write something like x=x* and y=epsilon y* where epsilon <<1 and plug this into your equations. You would then be able to see the relative sizes of terms with epsilons, epsilon^2 etc and maybe discard the highest order terms in epsilon if appropriate.